论文标题
均质$ 2 $ - 低尺寸的超曲面类型的非重音cr歧管
Homogeneous $2$-nondegenerate CR manifolds of hypersurface type in low dimensions
论文作者
论文摘要
在最近的一篇论文中，作者和I. Zelenko介绍了修改后的CR符号的概念，该符号用于组织$ 2 $ nondementementer的CR结构的本地不变性。 In this paper, we consider homogeneous hypersurfaces in $\mathbb{C}^4$, a natural frontier in the CR hypersurface Erlangen programs, and classify up to local equivalence the locally homogeneous $2$-nondegenerate hypersufaces in $\mathbb{C}^4$ whose symmetry group dimension is maximal among all such structures with the same local invariants用各自修改的符号编码。在考虑的维度中，我们表明，在具有给定修改的CR符号的均匀结构中，最对称的结构（称为模型结构）是唯一的。然后，通过在$ \ mathbb {c}^4 $中分类（最高局部等价）九个模型结构中，通过对均质超曲面的修改符号进行分类而间接实现分类。然后，应用用于获得此分类的方法以在较高维空间中找到同质性超曲面。总共$ 20 $本地非当量的最大对称同质性$ 2 $ 2 $ - 非字母超级曲面在$ \ mathbb {C}^5 $中描述，$ 40 $ $ 40 $此类hypersurfaces在$ \ mathbb {c}^6 $中描述了其中一些工程，其中一些工作是在其他工程中描述的。最后，描述了两个新序列，由$ n $索引，均为$ 2 $ 2 $ - nondementer hypersurfaces $ \ mathbb {c}^{n+1} $。值得注意的是，这些后一个序列之一的所有示例都可以实现为尼尔氏谎言组上的剩余结构。
In a recent paper, the author and I. Zelenko introduce the concept of modified CR symbols for organizing local invariants of $2$-nondegenerate CR structures. In this paper, we consider homogeneous hypersurfaces in $\mathbb{C}^4$, a natural frontier in the CR hypersurface Erlangen programs, and classify up to local equivalence the locally homogeneous $2$-nondegenerate hypersufaces in $\mathbb{C}^4$ whose symmetry group dimension is maximal among all such structures with the same local invariants encoded in their respective modified symbols. In the considered dimension, we show that among homogeneous structures with given modified CR symbols, the most symmetric structures (termed model structures) are unique. The classification is then achieved indirectly through classifying the modified symbols of homogeneous hypersurfaces in $\mathbb{C}^4$, obtaining (up to local equivalence) nine model structures. The methods used to obtain this classification are then applied to find homogeneous hypersurfaces in higher dimensional spaces. In total $20$ locally non-equivalent maximally symmetric homogeneous $2$-nondegenerate hypersurfaces are described in $\mathbb{C}^5$, and $40$ such hypersurfaces are described in $\mathbb{C}^6$, of which some have been described in other works while many are new. Lastly, two new sequences, indexed by $n$, of homogeneous $2$-nondegenerate hypersurfaces in $\mathbb{C}^{n+1}$ are described. Notably, all examples from one of these latter sequences can be realized as left-invariant structures on nilpotent Lie groups.
